Before proving the main properties of the map \(\widetilde{\operatorname{trop}}_{J_{g,n}^{log}}\)  in the following Theorem, recall [1]} that for any logarithmic stack \(\mathcal {X}\)  (fine, saturated and locally of finite type over a base field \(k\) ) there exists an Artin fan \(\mathcal {A}_{\mathcal {X}}\)  with faithful monodromy together with a natural strict morphism of logarithmic algebraic stacks (that we like to call the functorial logarithmic tropicalization morphism of \(\mathcal {X}\) )
\(\operatorname{trop}_{\mathcal {X}}:\mathcal {X}\longrightarrow \mathcal {A}_{\mathcal {X}}\) 
